Exponential stability of a free boundary problem with spherical symmetry for a gas bubble immersed in a bounded incompressible liquid

TL;DR

This paper analyzes the stability of spherical gas bubbles in a viscous liquid, proving exponential stability of equilibrium states and deriving decay rates, with implications for sonoluminescence and bubble dynamics.

Contribution

It establishes the exponential stability of spherical equilibria in a free boundary problem, characterizes equilibria via polynomial roots, and constructs a global center manifold for the model.

Findings

Equilibria are uniquely characterized by roots of a ninth-degree polynomial.

Spherical equilibria exhibit nonlinear and exponential asymptotic stability.

Decay rates depend on gas mass and temperature, with faster convergence for smaller gas mass or higher temperature.

Abstract

This paper is mainly concerned with the free boundary problem for an approximate model (for example, arising from the study of sonoluminescence) of a gas bubble of finite mass enclosed within a bounded incompressible viscous liquid, accounting for surface tensions at both the gas-liquid interface and the external free surface of the entire gas-liquid region. It is found that any regular spherically symmetric steady-state solution is characterized by a positive root of a ninth-degree polynomial for which the existence and uniqueness are proved and a one-to-one correspondence between equilibria and pairs of gas mass and liquid volume is established. We prove that these equilibria exhibit nonlinear and exponential asymptotic stability under small perturbations that conserve gas mass and liquid volume, and an equilibrium solution acts as a local minimizer of the energy functional, even under relatively large perturbations, with the proportionality constant determined by the adiabatic constant. Moreover, we construct a global center manifold to apply the center manifold theory. Our results apply to gases and liquids of all sizes. Furthermore, we derive the optimal exponential decay rate for small liquid volumes by analyzing the spectrum bounds of the associated linear operator and show that decreasing the gas mass or increasing the temperature can accelerate the convergence rate, a behavior not seen in unbounded liquid scenarios.